Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.6, pp. 291-292

Section Optical elements

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Optical elements

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The equation of Caglioti et al. (1958[link]), modified by Rietveld (1969[link]) and originally developed for constant-wavelength neutron diffraction, is frequently employed for parameterization of the instrumental profile. The FWHM and the pV mixing parameter η (replacing the Lorentzian and Gaussian widths of the Voigt) are then parameterized according to functions in tan(θ) and θ, respectively (Caglioti et al., 1958[link]; Leoni et al., 1998[link]; Scardi & Leoni, 1999[link]), [\eqalignno{{\rm FWHM}^2 &= U\tan^2\theta + V\tan \theta + W, &(3.6.18)\cr \eta &= a + b\theta + c\theta ^2.&(3.6.19)}]The parameters of the Fourier transform of a Voigt or pseudo-Voigt are then constrained to those of equations (3.6.18)[link] and (3.6.19)[link]. This is particularly convenient, as the Fourier transform of a Voigtian is analytical. In fact, for the pV case we have[T_{\rm pV}^{\rm IP} (L) = (1 - k) \exp (-\pi^2\sigma^2L^2/\ln 2) + k\exp (- 2\pi \sigma L), \eqno (3.6.20)]where σ = FWHM/2 and where (Langford & Louër, 1982[link]; Scardi & Leoni, 1999[link])[k = \left[1 + (1 - \eta)/\left(\eta \sqrt {\pi \ln 2}\right)\right]^{ - 1}. \eqno (3.6.21)]Equation (3.6.18)[link] can be found in the literature in a different form and with additional terms accounting, for example, for size effects: besides forcing a symmetry of the profile in 2θ space, these extra terms are a contradiction as they have nothing to do with the instrument itself.


Caglioti, G., Paoletti, A. & Ricci, F. P. (1958). Choice of collimator for a crystal spectrometer for neutron diffraction. Nucl. Instrum. 3, 223–228.Google Scholar
Langford, J. I. & Louër, D. (1982). Diffraction line profiles and Scherrer constants for materials with cylindrical crystallites. J. Appl. Cryst. 15, 20–26.Google Scholar
Leoni, M., Scardi, P. & Langford, J. I. (1998). Characterization of standard reference materials for obtaining instrumental line profiles. Powder Diffr. 13, 210–215.Google Scholar
Rietveld, H. M. (1969). A profile refinement method for nuclear and magnetic structures. J. Appl. Cryst. 2, 65–71.Google Scholar
Scardi, P. & Leoni, M. (1999). Fourier modelling of the anisotropic line broadening of X-ray diffraction profiles due to line and plane lattice defects. J. Appl. Cryst. 32, 671–682.Google Scholar

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